Finite Group Databasebeta

← #29671586

Subgroups — #29671586

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458803C211yesmaximal, characteristic
2Q8×#11758199C211yesmaximal, characteristic
3#21458104C211yesmaximal, characteristic, complemented
4#21445415C211yesmaximal, characteristic, complemented
5#21445366C211yesmaximal, characteristic, complemented
6#21445363C211yesmaximal, characteristic, complemented
7#21445211C211yesmaximal, characteristic, complemented
8#21445187C211yesmaximal, characteristic, complemented
9#21445089C211yesmaximal, characteristic, complemented
10#21445087C211yesmaximal, characteristic, complemented
11#21445082C211yesmaximal, characteristic, complemented
12#21445081C211yesmaximal, characteristic, complemented
13#21444702C211yesmaximal, characteristic
14Q16×#92418C211yesmaximal, characteristic, complemented
15C61×#3328C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17Q8×D244C2211yescharacteristic
18#16840165—21no—
19#16840165C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840144—21no—
23#16840141C2211yescharacteristic
24#16840113C2211yescharacteristic
25#16840105—21no—
26#16840047C2211yescharacteristic
27#16840042—21no—
28#16840042C2211yescharacteristic
29#16840039—21no—
30#16840039C2211yescharacteristic
31#16840019—21no—
32#16839984—21no—
33#16839981—21no—
34#16839972C2211yescharacteristic
35#16839967C2211yescharacteristic
36#16839964C2211yescharacteristic
37#16839916—21no—
38Q8×#92418C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C4×#11758202—21no—
41C4×#11758199—21no—
42C4×#11758199C2211yescharacteristic
43#16839841C2211yescharacteristic
44#16839769—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839509C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#556C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#485—21no—
60C61×#475C2211yescharacteristic
61C61×#470—21no—
62C61×#462C2211yescharacteristic
63C61×#451—21no—
64C4×Q16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#419—21no—
67C2×Q8×D61—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758199D411yescharacteristic
74C2×#11758189—21no—
75#14519777—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519772—41no—
79#14519765—21no—
80#14519757—21no—
81#14519757—41no—
82#14519753—21no—
83C4×Dic122—21no—
84C4×D244—21no—
85C4×D244C2311yescharacteristic
86#14519735—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710D411yescharacteristic
92#14519708—21no—
93#14519708—41no—
94#14519708—41no—
95C4⋊#92418—21no—
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519702—21no—
101#14519702—21no—
102#14519701—21no—
103#14519701—21no—
104#14519700—21no—
105#14519700—21no—
106#14519698—21no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697D411yescharacteristic
110#14519689—21no—
111C4×#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C2×Q16×C61—21no—
117C2×Q16×C61D411yescharacteristic
118C2×M16×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#126×C61—41no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—42no—
148C2×Q8×C61—81no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—41no—
154Q8×D61—42no—
155Q8×D61—81no—
156C2×D244C2×D411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—41no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—41no—
162Q16×C61—22no—
163Q16×C61—81no—
164M16×C61—81no—
165C2×C8×C61—21no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C22⋊C4×C61—21no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—41no—
186C42×C61C2411yescharacteristic
187#11758202—42no—
188#11758202—81no—
189#11758202—81no—
190#11758201—81no—
191#11758200—81no—
192#11758199—22no—
193#11758198—21no—
194#11758198—41no—
195#11758198—41no—
196#11758198—41no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—41no—
204C4×Dic61—41no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211#11758189—81no—
212C23×C61—21no—
213C22×D61—21no—
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—82no—
222Q8×C61#13512yes—
223D4×C61—21no—
224D4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—82no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241D244—21no—
242D244—21no—
243C4×D61—41no—
244C4×D61—41no—
245C4×D61—82no—
246Dic122—41no—
247Dic122—41no—
248Dic122—41no—
249Dic122—41no—
250Dic122—82no—
251C8×C61—21no—
252C8×C61—21no—
253C8×C61—41no—
254#92418—22no—
255#92418—21no—
256#92418—41no—
257#92418—41no—
258#92418—41no—
259#92418#13712yes—
260#19907—611nomaximal, Sylow
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61D4211yescharacteristic
265C2×D61—41no—
266C2×D61—41no—
267C2×D61—41no—
268C4×C61—22no—
269C4×C61—21no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61C2×#13511yescharacteristic
277C4×C61C2×#13711yescharacteristic
278C4×C61#57611yescharacteristic
279C4×C61#57711yescharacteristic
280Dic61—41no—
281Dic61—81no—
282Dic61—81no—
283#3393—611no—
284Q8×D8—611no—
285#3328—611no—
286#3328D6111yescharacteristic, complemented
287#1707—611no—
288#1699—611no—
289#1699—611no—
290#1674—611no—
291#1670—611no—
292#1652—611no—
293#1652—611no—
294#1651—611no—
295#1651—611no—
296#1589—611no—
297C8×Q16—611no—
298C2×C61—81no—
299C2×C61#327811yescharacteristic
300C2×C61#328011yescharacteristic
301C2×C61#329311yescharacteristic
302D61—81no—
303#556—611no—
304#556C2×D6111yescharacteristic
305D4×Q8—611no—
306D4×Q8C2×D6111yescharacteristic
307#500—611no—
308#500—1221no—
309#498—611no—
310#498—611no—
311#495—1221no—
312#494—611no—
313#487—611no—
314#487C2×D6111yescharacteristic
315#485—21no—
316#485—1221no—
317#475—611no—
318#475C2×D6111yescharacteristic
319#474—611no—
320#474—611no—
321#474—1221no—
322#474—1221no—
323#470—21no—
324#470—1221no—
325#464—1221no—
326#464—1221no—
327#462—611no—
328#462C2×D6111yescharacteristic
329#461—611no—
330#461—611no—
331#451—21no—
332#451—1221no—
333C8×Q8—611no—
334C8×Q8—611no—
335C4×Q16—1221no—
336C4×Q16C2×D6111yescharacteristic
337C4×D8—611no—
338C4×D8—1221no—
339#436—611no—
340#436C2×D6111yescharacteristic
341#419—21no—
342#419—1221no—
343#335—611no—
344#335—611no—
345#332—611no—
346#332—611no—
347#332—611no—
348#332—611no—
349Q8⋊C8—611no—
350Q8⋊C8—611no—
351D4⋊C8—611no—
352D4⋊C8—611no—
353C82—611no—
354C61#1990711yescharacteristic, complemented, Sylow
355C22×Q8—21no—
356C22×Q8—1221no—
357C2×Q16—21no—
358C2×Q16—1221no—
359C2×Q16—1221no—
360C2×Q16—1221no—
361C2×Q16C22⋊D6111yescharacteristic
362C2×SD16—1221no—
363C2×SD16—1221no—
364C2×D8—611no—
365C2×M16—21no—
366C2×M16—1221no—
367C4⋊Q8—21no—
368C4⋊Q8—611no—
369C4⋊Q8—611no—
370C4⋊Q8—1221no—
371C4⋊Q8C22×D6111yescharacteristic
372C4⋊Q8C22×D6111yescharacteristic
373#126—41no—
374#126—2441no—
375#124—21no—
376#124—1221no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—41no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8—2441no—
383C4×Q8—21no—
384C4×Q8—1221no—
385C4×Q8C22×D6111yescharacteristic
386C4×Q8C22×D6111yescharacteristic
387C4×D4—21no—
388C4×D4—611no—
389C4×D4—1221no—
390C4×D4C22×D6111yescharacteristic
391#117—21no—
392#117—1221no—
393#107—611no—
394#107—1221no—
395#107—1221no—
396#107—1221no—
397#107—1221no—
398#107C22⋊D6111yescharacteristic
399#106—21no—
400#106—1221no—
401#106—2441no—
402#106—2441no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—1221no—
407C4⋊C8—1221no—
408C4⋊C8C22×D6111yescharacteristic
409#103—21no—
410#103—21no—
411#103—21no—
412#103—21no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#103C22⋊D6111yescharacteristic
420#103C22⋊D6111yescharacteristic
421#102—611no—
422#102—611no—
423#102—1221no—
424#102—1221no—
425C22⋊C8—21no—
426C22⋊C8—1221no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8—1221no—
430C4×C8—1221no—
431C4×C8C22×D6111yescharacteristic
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—41no—
435C2×Q8—42no—
436C2×Q8—81no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8—4881no—
440C2×Q8—2441no—
441C2×Q8—2442no—
442C2×Q8C2×C22⋊D6111yescharacteristic
443C2×Q8C2×C22⋊D6111yescharacteristic
444C2×D4—611no—
445C2×D4D4×D6111yescharacteristic
446C22×C4—21no—
447C22×C4—41no—
448C22×C4—1221no—
449C22×C4—2441no—
450Q16—22no—
451Q16—81no—
452Q16—4881no—
453Q16—4881no—
454Q16—2442no—
455SD16—4881no—
456SD16—4881no—
457D8—1222no—
458M16—81no—
459M16—4881no—
460C2×C8—21no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8C2×C22⋊D6111yescharacteristic
468C2×C8C2×C22⋊D6111yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—611no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—2441no—
483C4⋊C4—2441no—
484C4⋊C4D4×D6111yescharacteristic
485C4⋊C4C2×C22⋊D6111yescharacteristic
486C4⋊C4C2×C22⋊D6111yescharacteristic
487C22⋊C4—21no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—1221no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C22⋊C4—2441no—
495C42—21no—
496C42—21no—
497C42—41no—
498C42—2441no—
499C42C23×D6111yescharacteristic
500C23—21no—
501C23—1221no—
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—82no—
510Q8—4882no—
511Q8—2441no—
512Q8—2441no—
513Q8—2441no—
514Q8—2441no—
515Q8#1451979712yes—
516D4—21no—
517D4—21no—
518D4—1221no—
519D4—1221no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—82no—
530C2×C4—1221no—
531C2×C4—4882no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4C22×C22⋊D6111yescharacteristic
538C2×C4#1451982211yescharacteristic
539C8—21no—
540C8—21no—
541C8—41no—
542C8—612no—
543C8—1222no—
544C8—1221no—
545C8—2441no—
546C8—2441no—
547C8—2441no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—2441no—
552C22—2441no—
553C22—2441no—
554C22D4×C22⋊D6111yescharacteristic
555C4—22no—
556C4—21no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—4881no—
564C4—4881no—
565C4—2441no—
566C4C2×#1451979711yescharacteristic
567C4#137×D6111yescharacteristic
568C4#1684056511yescharacteristic
569C4#1684057111yescharacteristic
570C2—81no—
571C2—4881no—
572C2#2145747911yescharacteristic
573C2#2145750011yescharacteristic
574C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3328C2
socleC22×C61D42
prime radicalC2×#11758199D4
hypercenter#3328D61
hypocenterC61#19907
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3328

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #21458104C2semidirect
4. #21445415C2semidirect
5. #21445366C2semidirect
6. #21445363C2semidirect
7. #21445211C2semidirect
8. #21445187C2semidirect
9. #21445089C2semidirect
10. #21445087C2semidirect
11. #21445082C2semidirect
12. #21445081C2semidirect
14. Q16×#92418C2semidirect
15. C61×#3328C2semidirect
286. #3328D61semidirect
354. C61#19907semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214588032
Q8×#117581992
#214581042
#214454152
#214453662
#214453632
#214452112
#214451872
#214450892
#214450872
#214450822
#214450812
#214447022
Q16×#924182
C61×#33282
#1990761